Spectral Properties of Fourth Order Differential Operators with Periodic and Antiperiodic Boundary Conditions

Abstract

In this paper, we consider the following periodic and antiperiodic problem yiv + p2 (x) y + p1 (x) y + p0 (x) y = λy, 0 <x< 1, y(s) (1) − (−1)σ y(s) (0) = 0, s = 0, 3, where λ is a spectral parameter; pj (x) ∈ L1(0, 1), j = 0, 1, p2(x) ∈ W1 1 (0, 1) with 1 0 p2(ξ)dξ = 0 are complex-valued functions and σ = 0, 1. The boundary conditions of this problem are periodic-antiperiodic boundary conditions and it is well known that they are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. Under the condition (p2 (1) − p2 (0) − 2c1) (p2 (1) − p2 (0) + 2c1) = 0, it is proved that all the eigenvalues (except for finite number) are simple, where c1 = 1 0 p1(ξ)dξ. Furthermore, we prove that the system of root functions of this spectral problem forms a basis in the space Lp(0, 1), 1 <p< ∞, when p1(1) = p1(0); p(s) 2 (1) = p(s) 2 (0), s = 0, 1; pj (x) ∈ Wj 1 (0, 1), j = 0, 1, 2; c1 = 0. Also, it is shown that this basis is unconditional for p = 2.